14856
J. Phys. Chem. 1996, 100, 14856-14864
ARTICLES
Properties of Silicon Nanoparticles: A Molecular Dynamics Study
Michael R. Zachariah* and Michael J. Carrier
National Institute of Standards and Technology, Gaithersburg, Maryland 20899
Estela Blaisten-Barojas
Institute for Computational Sciences and Informatics, George Mason UniVersity, Fairfax, Virginia 22030
ReceiVed: December 20, 1995; In Final Form: March 12, 1996X
Constant energy molecular dynamics simulations of silicon cluster growth have been conducted for clusters
up to 480 atoms using the Stillinger-Weber empirical interatomic potential. It is found that the interior
atoms of the 480-atom clusters, at the temperatures used, show bulklike characteristics. The cluster binding
energy has been fit to an expression that separates the surface and bulk contributions to the energy over wide
temperatures and size ranges. The average surface energy of an atom was found to be independent of cluster
size and of a magnitude relative to the bulk, such that all cluster sizes were stable under the conditions
studied here (600 < T < 2000 K). The photon density of states is similar to bulk silicon and does not show
a strong cluster size dependence. Atomic self-diffusion coefficients have been calculated and compare quite
well with experimental data on self-diffusion coefficient measurements of silicon surfaces.
Computational Method
Introduction
Nanometer processing is receiving considerable interest from
a variety of communities, including those from microelectronics
and advanced materials. One of the challenges in these areas
is the processing of very fine particles. This would include
their controlled growth, chemical reactivity, and transport
properties. Considerable attention has been paid to the growth
of particles in the range of 100 nm and up; however, process
modelers have implicitly assumed that small particle growth is
unimportant. These issues have however reemerged due to the
interest in nanometer particle processing.1 Understanding the
underlying physics and chemistry is necessary for the construction and application of robust phenomenological models for
describing particle formation.2 This is important from the
perspective of both the microelectronics community which hopes
to minimize particle growth in the vapor during chemical vapor
deposition and the ceramics community which hopes to develop
the ability to grow from the vapor, spherical, unagglomerated
nanometer scale particles.3-5 Questions regarding the nature
of cluster growth kinetics and morphology are of concern.
While a large body of literature exists on calculations of the
properties of bulk silicon, a much smaller amount of work has
been directed toward silicon clusters containing hundreds of
atoms. In the cluster studies, the majority of the research
involves silicon clusters with 20 atoms and fewer6-20 with the
remainder extending the range to about 50 atoms,21-29 with no
attempt to extend the sizes to what could be called nanoscale
materials. In this work we extend the size range of calculation
to much larger (N ) 480) clusters and treat a wide range of
temperatures (600 < T < 2000 K).
* To whom correspondence should be addressed. E-mail: mrz@
tiber.nist.gov.
X Abstract published in AdVance ACS Abstracts, July 15, 1996.
The approach used in this work is to apply an atomistic
simulation using classical molecular dynamics (MD) methods.21,22 The molecular dynamics calculations were conducted
by solving the equations of motion with the velocity form of
the Verlet algorithm,30 with a time step of 5.7 × 10-4 ps to
ensure energy conservation. Computations were conducted
using the silicon potential proposed by Stillinger and Weber
(SW).31 This empirical potential contains two- and three-body
interactions which take into consideration the directional
characteristic of the covalent bonding and are given in
eqs 1a-1c.
V ) ∑ V2(i,j) +
i,j
i<j
∑
i,j,k
V3(i,j,k)
(1a)
i<j<k
two-body term
V2(r) ) A(Br-p - r-q) exp[(r - a)-1], r < a
) 0, r g a
(1b)
three-body term
V3(ri,rj,rk) ) h(rij,rik,θjik) + h(rji,rjk,θijk) + h(rki,rkj,θikj)
h(rij,rik,θjik) ) λ exp[γ(rij - a)-1 +
γ(rik - a)-1] cos(θjik + 1/3)2 (1c)
where r is the distance between a pair of atoms, a is the cutoff
distance of the two-body potential, and θijk is the bond angle
between a triplet of atoms. The parameters (A, B, p, q, λ, γ)
are constants formulated by Stillinger and Weber and are specific
to liquid silicon.31 While this potential does have some
limitations, it is among the best in the literature for representing
high-temperature properties of silicon.12 The diamond crystal
S0022-3654(95)03773-7
This article not subject to U.S. Copyright. Published 1996 by the American Chemical Society
Properties of Silicon Nanoparticles
structure of silicon may be thought of as a network of bonds
due to “many-body” effects that a pairwise interatomic potential
cannot realistically reproduce. While other potentials are
available to simulate silicon, most have been developed for
specific purposes, such as accurately reproducing the geometries
of very small clusters or specific surface faces, and are generally
not as thoroughly tested as SW. The SW potential was chosen
because it accurately predicts the elastic constants and cohesive
energy of the diamond structure of silicon. It also does well in
predicting the bulk melting characteristics and structure of liquid
silicon and is one of the most extensively applied silicon
potentials for large-scale systems. The SW potential has also
been shown to yield a good description of phonon frequencies,
as well as surface epitaxial growth at high and intermediate
temperatures. While it is well-known that the SW potential does
not give accurate structures for very small (<15 atoms) Si
clusters at 0 K,14,15 it does get the correct structure for bulk
liquid Si.12 Because most synthesis processes leading to cluster
formation occur at high temperatures, it was felt that liquidlike
characteristics should play a dominant role in any description
of cluster growth. Our objective is to look at particles created
in high-temperature processes which are either liquid or at nearly
melted conditions. Since the SW potential has been shown to
be useful in studies of bulk liquid, and because our interest lies
more toward higher temperatures and increasingly larger cluster
sizes, we believe there is justification in the claim that SW
results for clusters improve as they get larger and hotter. We
are using unterminated clusters in this work for two reasons.
First, we are currently investigating the formation of silicon
nanoparticles via reaction of Na with SiCl4 where no hydrogen
is involved. Second, this is a prelude to an investigation on
the coalescence behavior of silicon clusters where it is hoped
that extracting phenomenological relationships about particle
sintering will be simpler in a single-component system. Furthermore, the work can be related more easily to other metal
systems.
The simulations are conducted in two stages:
1. Cluster Preparation/Equilibration. Liquid cluster configurations of 30, 60, 120, 240, and 480 atoms were generated
from a related study of cluster-cluster collisions. These were
used as a starting point for the creation of the clusters used in
this work. The first step in the equilibration process was to
remove any angular momentum from the cluster. On average,
however, the angular momentum only contributed approximately
5 K to the temperature of the cluster. The cluster was then set
to the desired temperature using constant temperature molecular
dynamics for a period of 50 ps. At the end of this time segment
the simulation was switched to constant energy for 20 ps, and
the temperature was recorded over that interval. If the average
temperature of the cluster did not change more than 10 K over
this period, the equilibration was considered a success; otherwise, the process was repeated.
2. Cluster Property Simulation. Following equilibration,
cluster properties were averaged using constant energy molecular
dynamics over a period of 25 ps.
Cluster Structure: Surface vs Bulk. Before we embark on
an analysis of the internal energy and properties of these clusters,
we define more clearly the exact structural nature of the clusters
prepared here. The shapes of the clusters used in this work are
all spherical as this is the equilibrium shape for clusters over
28 atoms.25 As an initial look at the structure of these clusters,
we calculated the radial distribution function g(r) for the largest
clusters (480 atoms) across our temperature range. This was
used to obtain the mean bond length from the position of first
peak and the maximum bond stretch (nearest-neighbor cutoff)
J. Phys. Chem., Vol. 100, No. 36, 1996 14857
Figure 1. Pair correlation function of the 480-atom cluster at T )
2060 K for (a) all 480 atoms, (b) the inner 60 core atoms, and (c) the
outer 60 surface atoms.
from the location of the first minimum. For the hottest clusters
at 2060 K there are two minima of about equal depth between
the first and second peak instead of one (Figure 1a). The
locations of the two minima at 0.306 and 0.343 nm are in
agreement with those obtained for bulk solid and bulk liquid
silicon, respectively.31 To test whether this finding was related
to bulk and surface effects, the radial distribution function was
calculated for both the 60 innermost (Figure 1b) and the 60
outermost atoms of the cluster (Figure 1c). For the inner core
atoms, the first of the two minima has been replaced by a
shoulder on the right-hand side of the first peak, leaving a first
minimum at 0.345 nm similar to those calculated by Stillenger
and Weber in molten silicon at 0.340 nm (2054 K)31 and by
Kluge et al. at 0.335 nm (1850 K).32 For the surface atoms,
the structure displays the characteristic second peak of the
underlying liquid and resembles an amorphous material. Clusters at 650 K show a better resolved g(r) (Figure 2a). However,
there is little difference in the g(r) between the core (Figure
2b) and surface atoms (Figure 2c), and both display a clear
amorphous structure.
Differences between core and surface atoms are also seen in
the calculation of the coordination number. The coordination
number is determined from the nearest-neighbor cutoff distance
obtained from the g(r) and averaged over the number of
neighbors each atom has within that radius. Figure 3 shows
that the surface atoms are between 3- and 4-coordinated over
the temperature range studied while the inner atoms are close
to 5-coordinated at temperatures up to 1600 K. The 2060 K
cluster shows a jump in coordination for the core atoms to 8.49,
which is reasonably close to the value of 8.07 obtained by
Stillinger and Weber for bulk molten silicon at 2054 K31 and
indicates that the hottest clusters are liquidlike. The mean bond
length increased with temperature as expected but also showed
14858 J. Phys. Chem., Vol. 100, No. 36, 1996
Zachariah et al.
Figure 4. Mean bond angle for a 480-atom cluster as a function of
temperature for all 480 atoms, the inner 60 core atoms, and the outer
60 surface atoms.
TABLE 1: Coordination Distribution As Calculated by
Luedtke and Landman for Bulk Silicon33,34
coordination
Figure 2. Pair correlation function of the 480-atom cluster at T )
650 K for (a) all 480 atoms, (b) the inner 60 core atoms, and (c) the
outer 60 surface atoms.
amorphous
(352 K)
glass
(352 K)
crystal
(352 K)
liquid
(1665 K)
3
0.005
0.002
0
0.023
4
0.878
0.410
1
0.294
5
0.115
0.522
0
0.477
6
0.003
0.066
0
0.185
7
0
0.001
0
0.021
av coordn
4.66
4.00
4.89
neighbor
0.2933
cutoff, nm
mean bond
108.3 ( 14.7 106.6 ( 22.6 109.4 ( 2.7 103.9 ( 26.8
angle, deg
TABLE 2: Coordination Distribution Calculated in This
Work for the 60-Atom Inner Core of a 480-Atom Cluster
Figure 3. Average coordination number for a 480-atom cluster as a
function of temperature for all 480 atoms, the inner 60 core atoms,
and the outer 60 surface atoms.
differences between surface and core atoms with the core atoms
having a larger bond length to accommodate higher coordination
numbers. The latter result was consistent with the mean bond
angle shown in Figure 4 which showed that surface atoms had
bond angles closer to tetrahedral (109°). As the clusters are
heated, the bond angles decrease in order to accommodate
increased coordination.
Cluster Structure: Composition. Due to the large temperature range these simulations span, clusters can exist in one of
the following states: liquid, glass, crystal, or amorphous. In
classifying the clusters, we looked at several quantities: the
radial distribution function, bond angle distribution, and mean
coordination
650 K
1560 K
2060 K
3
4
5
6
7
8
9
10
11
12
av coordn
neighbor cutoff, nm
mean bond angle, deg
0.006
0.438
0.492
0.062
0.002
0
0
0
0
0
4.62
0.291
106.8
0.002
0.127
0.417
0.341
0.099
0.012
0
0
0
0
5.45
0.301
102.59
0
3.8 × 10-4
0.009
0.060
0.166
0.276
0.258
0.154
0.060
0.014
8.49
0.345
97.31
coordination number distribution. These were all derived from
atoms at the core of the cluster to avoid the effects that the
surface atoms might have on these quantities and enable a
comparison with results of bulk material. Luedtke and Landman33,34 simulated bulk silicon using the SW potential and
classified the states of the material by coordination number
distribution and mean bond angle (Table 1). In their calculations, amorphous, glassy, and crystal materials were prepared
at 352 K, while the liquid was characterized at the melting point
(1664 K). The crystal state was characterized as every atom
having four bonds, while amorphous material had about 10%
of five bonded atoms and the “glass” is about 50% 5-coordinate.
The liquid state at the melting point shows similar behavior to
the glass in that the majority of atoms are 5-coordinate.
Comparing the Luedtke-Landman (L-L) results as presented
in Table 1 with our results for the 60 interior atoms of the 480atom clusters in Table 2 shows good agreement between the
“bulk” atoms of our 650 K cluster and the bulk L-L “glassy”
Properties of Silicon Nanoparticles
J. Phys. Chem., Vol. 100, No. 36, 1996 14859
Figure 5. Bond angle distribution for the inner 60 atoms of a 480-atom cluster for (a) 2060, (b) 1560, and (c) 650 K. Coordination distribution
for the inner 60 atoms of a 480-atom cluster for (d) 2060, (e) 1560, and (f) 650 K.
solid. The formation of a “glassy” or supercooled liquid cluster
rather than amorphous material was expected because of the
inability to quench liquid silicon to an amorphous state using
the SW potential as reported by Broughton and Li.35 While
there is a reasonable match between our 1560 K cluster and
the L-L material at the melting point (1665 K), the cluster at
2060 K is far into the liquid state and the coordination
distribution matches that reported by Stillenger and Weber for
bulk liquid.31
These high-temperature structural trends are more clearly
revealed in Figure 5a-f which indicate the evolution of the bond
angle distribution and the coordination number with increasing
temperature. Bond angle distributions shown in Figure 5a-c
indicate that as the temperature increases, smaller bond angles
are increasingly favored, consistent with the increased coordination number as the material becomes more compact. The
coordination distribution for the 2060 K cluster shows that the
majority of the atoms are 8- and 9-coordinated with a mean of
8.49, in reasonable agreement with a mean coordination number
of 8.07 calculated by Stillinger and Weber31 at 2062 K. Kluge
et al.32 found an average coordination number of 7.7 with an
1850 K liquid. They also found only 0.5% of the atoms at this
temperature had four bonds, in agreement with our calculations
where only 0.04% of the atoms were 4-coordinated.
Since the melting point is known to vary with cluster size,
for simplicity of terminology, when we refer to solid and liquid
clusters, “solid” will refer to the ∼600 K glassy or supercooled
liquid, while “liquid” will refer to the molten ∼2000 K material
which is in fact several hundred degrees above the melting point.
For tetrahedrally bonded materials such as silicon, the atoms
on the surface have a coordination number of 3 if no surface
reconstruction takes place. Figure 6a shows that the outer 60
Figure 6. Coordination distribution for the outer 60 atoms of a 480atom cluster for (a) 650 and (b) 2060 K.
surface atoms of our 650 K, 480-atom cluster has 81%
3-coordinate and 19% 4-coordinate atoms, giving us a mean
coordination number of 3.2. For liquidlike clusters we should
expect that the surface atoms will exhibit a coordination number
larger than 3 and presumably extensive location reaccommodation of the outer atoms. Figure 6b shows that, for the outer
60 surface atoms of the 2060 K, 480-atom cluster, the number
of 3-coordinated atoms has dropped to 43% and 4-coordinated
atoms doubled to 40%, giving a mean coordination number of
3.7.
14860 J. Phys. Chem., Vol. 100, No. 36, 1996
Figure 7. Average coordination number as a function of temperature
and cluster size.
Figure 8. Dependence of cluster binding energy per atom on cluster
size (N atoms) for five different temperatures. (Solid lines are fits from
eq 2.)
The average coordination numbers for all cluster sizes
considered in our simulations are shown in Figure 7. On
average there is a gradual increase in the coordination number
as temperature is increased for all clusters. The average
calculated coordination number for liquid clusters is considerably lower than the experimental value of 6.4 for the bulk36,37
due to the large number of surface atoms relative to bulk, while
solid clusters have effectively reached a 4-coordinate state by
240 atoms. The most sensitive effect on average coordination
number is in fact cluster size, which shows a rapid rise in
coordination number as the size of the cluster increases,
consistent with the rise in the percentage of bulk relative to
surface atoms as the cluster gets larger.
Equilibrium Cluster Energetics. Because of the large
surface-to-volume ratio, cluster properties vary with size.
Indeed, this is one of the factors that presumably make nanoscale
materials so interesting. The approach to bulk properties as
cluster size is increased is in general specific to the property of
interest. Figure 8 shows the effect of cluster size on binding
energy per atom for various temperatures. Increasing the size
of a spherical cluster results in a monotonically increasing cluster
binding energy in agreement with the work of Kaxiras and
Jackson.25 Freund and Bauer38 have discussed the relationship
between cluster binding energy and cluster size for metal clusters
and obtained a N-1/4 relationship for the surface contribution
to the binding energy per atom, based on a number of
independent experimental measurements. Northby39 has shown
that for Lennard-Jones clusters, which are close-packed, there
is a crossover from icosahedral configurations into fcc when
about 14 atomic shells surround a central atom. For these
classes of materials, surface atoms keep a coordination number
between 6 and 7 and as clusters grow in size the surface
Zachariah et al.
Figure 9. Binding energy per atom at 0 K versus cluster size (N).
Comparison of the predicted values in this work with binding energies
of spherical clusters in the perfect diamond structure (*).
contribution to the binding energy per atom tends to vanish as
N-1/3. In contrast, for network type materials (such as silicon),
atomic shells cannot be defined, but rather coordination shells
need to be identified. In particular, small silicon clusters of
fewer than 10 atoms are thought to be metallic in nature,13,14
while the bulk material is covalent.23 Several estimates have
been made to determine the transition from metallic to covalent
bonding and range from as few as 50 atoms23 to as many as
4000 atoms;13 however, none of these works describe the size
dependence of the energy contribution to the binding energy.
An analysis of data in Figure 8 allows us to propose a fit to
an expression showing the relative importance of bulk (volume)
and surface energy (surface area) contributions to the total
binding energy as follows
binding energy ) Eb + Es
(2)
where the bulk contribution Eb is negative and the surface
contribution Es is positive. If a cluster with N atoms is assumed
to be spherical, then the binding energy per atom is given by
binding energy/N ) b(T) + 4πrws2σ(T)N-1/3
(3)
where the bulk binding energy b and the surface tension σ
depend only on temperature. In eq 3, rws is the Wigner-Seitz
radius40 which for silicon is 1.68 Å, consistent with the lattice
constant a ) 5.43 Å of the diamond structure. We have
assumed rws to be independent of temperature. From our results
in Figure 9 we can extract a dependence of the surface tension
with temperature and extrapolate the value of the bulk binding
energy at zero temperature. Since this is a classical calculation,
the equipartition of energy should apply and thus you get a bulk
binding energy which is dependent only on temperature and
not cluster size
b(T) ) b(0) + CvT
(4)
where b(0) is the bulk binding energy at zero temperature and
Cv is the constant-volume heat capacity. The size independence
of the binding energy is shown in Figure 12 where you see that
the inner atoms all have the same binding energy for each of
the four cluster sizes. Below the melting temperature the
potential energy contributes Cv/R ) 3 as for any crystal. The
best temperature dependence of the surface tension was obtained
by utilizing an expression of the following form:
σ(T) ) σ0{1 + A(1 - T/B) exp[(T - Tm)/Tm]}
(5)
where σ0 is the surface tension at zero temperature, A ) 0.068,
Properties of Silicon Nanoparticles
J. Phys. Chem., Vol. 100, No. 36, 1996 14861
Figure 10. Comparison of surface to the bulk binding energy as a
function of cluster size (N is the number of atoms in the cluster) for
two temperatures.
Figure 11. Derivative of potential energy function per atom as a
function of cluster size.
TABLE 3: Predicted Values of Cluster Properties
property
b(0), kJ/mol
σ0, J/m2
T m, K
CV/R
Tc, K
calculated values,
this work
bulk values,
exptl
-396.7
0.985
1500
3.0 if T < Tm
4.1 if T > Tm
6350
-418
0.73 at T > 1713 K
1713
3.0 if T < Tm
3.2 if T > Tm
6400
and B ) 4000 K and the critical temperature is obtained when
σ(Tc) ) 0. From the fit to our computer experiment data of
Figure 9 using eqs 4 and 5, we obtained values for b(0), σ0,
Tm, Tc, and CV, which we list in Table 3. The value of b(0)
indicates that the material growth at these cluster sizes is not
perfectly tetrahedral, but rather resembles a glassy material.
Using eqs 3 and 4 we can predict the effects of size on the
binding energy at zero temperature. This calculation, is shown
in Figure 9 (solid line), and compares our predicted SW potential
results for the binding energies of rigid spherical clusters with
a diamond structure (asterisks). It is apparent that growth
following a nondiamond type is favored for clusters up to 2000atom, whereapon a crossover occurs favoring growth corresponding to that of bulk silicon. Actual growth conditions are
usually kinetically driven and may not follow the most
thermodynamically favored state.
The continuous lines in Figure 8 show the result of a best fit
of eq 3 using the parameters in Table 3. These results indicate
that the molecular dynamics simulations can be fit quite nicely
to such an expression. The bulk energy contribution to the
clusters total potential energy becomes more negative as the
cluster size is increased (increasing stability), while the surface
term shows a shallow positive increase. In Figure 10 the
absolute value of the ratio of surface to bulk energy at 600 and
2000 K obtained from eq 3 indicates the relative importance of
these two contributions as a function of both temperature and
cluster size. The results show that, in the temperature range
explored, increasing temperature causes an increase in the Es/
Eb. Increasing cluster size decreases the surface-to-volume ratio
and results in a decrease in the relative importance of the surface
energy. Under all conditions calculated here, the cluster binding
energy was negative which implies that the clusters were always
stable even at high temperatures (2000 K). Extrapolation of
eq 3 to cluster sizes fewer than about 30 atoms is strictly not
correct since it is well-known that SW does a poor job of
describing very small cluster geometries; however, from a
qualitative perspective, extrapolation to small sizes (Figure 10
insert) shows that the bulk term dominates over the surface term
even for a single silicon atom. The implications are that cluster
growth for temperatures up to at least 2000 K will not be limited
by a thermodynamic barrier.
The approach to bulk behavior is one of the most interesting
aspects to the understanding of nanostructured materials, which
encompasses mechanical, optical, and chemical properties. One
aspect of the approach to bulk behavior is the behavior of the
average binding energy per atom as a function of cluster size
as we showed in Figure 9. Alternatively, we can observe the
incremental change in the cluster binding energy as a function
of cluster size. This is shown in Figure 11 and indicates that
an asymptote is rapidly reached by a cluster of only about 200
atoms.
The potential energy of the cluster may be partitioned into a
site energy associated to each atom. The instantaneous atomic
binding energy as a function of an atom’s position relative to
the center of mass of the cluster provides a view of the relative
energies of surface and interior atoms. Figure 12 shows an
instantaneous snapshot of site energies for four cluster sizes at
600 K. These solid clusters show common features of a tightly
bound inner core enveloped by less strongly bound surface
atoms whose binding energy is approximately 80% of that for
the inner atoms. Most interesting is the fact that outer atoms
do not seem to show any cluster size dependence to their binding
energy. Indeed, one might expect that cluster properties which
are so dependent on size would be sensitive to this parameter.
The indications are that size dependence may be more a function
of the relative amount of bulk and surface atoms rather than
the exact nature of the surface, at least on the level of these
calculations where no electronic effects are accounted for.
Liquid clusters at 2000 K display a similar behavior to that of
the solid clusters shown in Figure 12, with the exception that
the distinction between inner and surface atoms is blurred. A
similar analysis for the instantaneous kinetic energy shows no
distinctive features as one progresses from the center to the outer
edge of the cluster, consistent with a good equilibration.
Figure 13 shows the instantaneous site energy broken down
by the site coordination number as a function of distance from
the center of mass for the 600 K, 480-atom cluster. Consistent
with the results of Figure 3, outer atoms have the lowest
coordination number and smallest bond length and are also the
atoms with the lowest binding energy. As one moves to the
interior of the cluster, the coordination number increases along
with the binding energy. Such an analysis is not so easily
constructed for the 2000 K clusters because of their high atomic
mobility.
Vibrational Energy. Vibrational energy transfer processes
are very important to the mechanism of cluster stabilization
during growth as well as its radiative properties. The vibrational
density of states was calculated by taking the FFT of the velocity
14862 J. Phys. Chem., Vol. 100, No. 36, 1996
Zachariah et al.
Figure 12. Instantaneous atomic binding energy as a function of position relative to the center of mass for four different cluster sizes at 600 K.
autocorrelation function C(t):
1
N
N
〈∑vi(t0)‚vi(t0+t)〉
(6)
i)1
Figure 14 shows the density of vibrational states of the 480atom cluster as a function of cluster temperature. A comparison
with the calculated phonon density of states for bulk silicon41
is also included in the insert of Figure 14. It is clear that the
density of vibrational states is not significantly altered as a result
of the large number of surface atoms. The spectra show the
two dominant modes found in bulk silicon (acoustic, 150 cm-1;
optical, 400 cm-1). As the cluster temperature is increased,
the phonon density displays broadening and a softening of the
modes to lower frequencies. The density of vibrational states
does not show a major discernible size dependence.
Silicon Self-Diffusion Coefficients. Self-diffusion coefficients have been calculated from the mean-square displacement
(MSD) from eq 7:
MSD )
1
N
∑[ri(t) - ri(0)]2
N i)1
MSD ) 6D/t
(7)
(8)
Figure 15 shows calculations of the MSD in a 480 atom-cluster
(at 600, 1200, and 2000 K) as a function of time. The curvature
in the MSD toward an asymptote results from the finite nature
of the cluster. The diffusion coefficient can be evaluated from
the initial slope of the MSD (eq 8).
Figure 16 shows the calculated diffusion coefficients as a
function of temperature and cluster size. The non-Arrhenius
behavior is suggestive of a phase change over the temperature
range sampled.
Because clusters of this size are essentially composed of
surface rather than volume type atoms, comparison with
experiment is most appropriate with surface self-diffusion data.
Unfortunately, such data are scarce and of questionable accuracy. The most direct measurement has been made by
Makowiecki and Holt,42 who reported a silicon surface diffusion
constant of 4 × 10-8 cm2/s at 1273 K. Our data for the largest
cluster (480 atoms) gave a diffusion constant of about 1 × 10-5
cm2/s at 1270 K, which is significantly larger than the
experimental results, indicating liquidlike behavior at lower
temperatures than bulk surfaces. Mo et al.43 did a high-quality
STM study at lower temperatures, which we have extrapolated
to the range of temperatures we have studied. The comparison
is quite good as is that for the Robertson44 work which crosses
our data, all be it with a different activation energy. Clearly
there exists substantial discrepancy within the bulk experimental
data, with no obvious way to access the accuracy of the results.
Certainly, it is well-known that the Stillinger-Weber potential
underpredicts the binding energy in bulk silicon which would
tend to increase the computed surface diffusion coefficient. On
the other hand, the tight binding molecular dynamics study of
Wang et al.45 for liquid silicon at 2000 K agrees quite well with
our calculated diffusion coefficient of the largest cluster used
(480 atoms) at the same temperature. The most obvious source
of error in the experiment, namely surface oxidation would tend
to decrease the diffusion coefficient, consistent with the available
data which either lie with our data or below it.
Conclusions
Classical molecular dynamics simulations using the threebody Stillinger-Weber interatomic potential have been used
to simulate the equilibrium characteristics of clusters over a wide
range of cluster sizes (30 < n < 480 atoms) and temperatures
(600 < T < 2000 K). The structure of the innermost 60 atoms
of 480 atom clusters over the entire temperature range were
found to resemble that of bulk “glassy” silicon. Three distinct
phases for these clusters were observed. At low temperatures
(650 K) the cluster was found to correspond to a “glass” or
Properties of Silicon Nanoparticles
J. Phys. Chem., Vol. 100, No. 36, 1996 14863
Figure 13. (a) Site energy and (b) mass distribution versus distance from center of mass for N ) 480 cluster at T ) 600 K. (c-f) Site energy as
a function of distance from center of mass for the atoms with coordination 3-6, respectively.
Figure 14. Phonon density of states for a 480-atom cluster as a function
of cluster temperature.
Figure 15. Mean-square displacement of atoms as a function of time.
supercooled liquid as observed by Ludetke and Landman,33,34
while at the melting point (1560 K) the cluster showed a
coordination distribution corresponding again to the Ludetke
and Landman study for bulk silicon at the melting point (1665
K). At still higher temperatures (>2000 K), cluster structures
corresponded quite closely to those calculated by Stillinger and
Weber for bulk liquid silicon at 2054 K.31
Cluster binding energy calculated as a function of size and
temperature have been fitted to an expression that accounts for
bulk and surface contributions. The cluster results as compared
14864 J. Phys. Chem., Vol. 100, No. 36, 1996
Figure 16. Arrhenius plot of self-diffusion coefficients versus inverse
of temperature.
with bulk properties are consistent. Growth seems to be through
glass-like rather than crystalline structures. Phonon density
spectra show no outstanding features when compared to the bulk
implying that thermal transport properties within the clusters
should be similar. Calculated self-diffusion coefficients show
that cluster results are similar to experimental data of bulk
surfaces at intermediate temperatures and the diffusion coefficients for large clusters at high temperatures agree with values
of a bulk liquid.
References and Notes
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